DIFFERENTIAL INEQUALITIES METHOD TO nTh-ORDER BOUNDARY VALUE PROBLEMS GUANGWA WANG, MINGRU ZHOU, AND LI SUN Received 31 March 2005; Revised 18 October 2005; Accepted 7 December 2005 By the theory of differential inequality, bounding function method, and the theory of topological degree, this paper presents the existence criterions of solutions for the general nth-order differential equations under nonlinear boundary conditions, and extends many existing results. Copyright © 2006 Guangwa Wang e t a l. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction From Nagumo [10], there have been many accomplishments on the study of the exis- tence of solutions for boundary value problems (BVPs) using the theory of differential inequality (cf. [1–9, 11–17]). However, for the nth-order nonlinear differential equations with the nonlinear boundary conditions, results are very few. The authors made some attempts to solve the nth-order Robin problem [14]. Now we are concerned with the nth-order nonlinear BVP: y (n) = f t, y, y , , y (n−1) , P i y(a), y (a), , y (n−1) (a) = 0, i = 1, ,n − 1, P n y(b), y (b), , y (n−1) (b) = 0, (1.1) where t ∈ I = [a,b], f (t,ξ 0 ,ξ 1 , , ξ n−1 ) ∈ C(I × R n ,R), P i (η 0 ,η 1 , , η n−1 ) ∈ C(R n ,R), P n (ζ 0 ,ζ 1 , , ζ n−1 ) ∈ C(R n ,R). Our method is not only modifying the nonlinear function in the original equations, but also transforming the original nonlinear boundary conditions into some new bound- ary conditions which are easy to discuss. Thus, we get the new BVP which will be dis- cussed firstly, then the judgement of the existence of solutions for the orig inal BVP will be attained naturally. This technique dealing with the nonlinear problem is simpler and Hindaw i Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 12040, Pages 1–12 DOI 10.1155/JIA/2006/12040 2Differential inequalities method to nth-order BVPs clearer compared with the method of shooting . However, it has scarcely been used in the available reference materials. The paper is organized as follows. In Section 2, we give out some basic concepts and the preparative theorem. In Section 3, the main result is presented and proved. In Section 4, a more general BVP is studied. Finally, in Section 5, we use the results to solve an example which cannot be solved by [1–17]. 2. Preparative theorem 2.1. Basic concepts. We first define a function δ(r,x,s) ≡ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ r if x<r, x if r ≤ x ≤ s, s if s<x, (2.1) where r,x,s ∈ R, r ≤ s. Definit ion 2.1. Assume that α(t), β(t) ∈ C n (I,R). The pair of functions (α(t),β(t)) is called a bounding function pair (or simply, a bounding pair) of BVP (1.1)incasethere exists N>0 such that for all u(t) ∈ C n (I,R): (i) α ( j) (t) ≤ β ( j) (t), t ∈ I, j = 0,1, ,n − 2; (ii) α (n) (t) ≥ f (t,u(t),u (t), ,u (n−3) (t),α (n−2) (t),α (n−1) (t)), β (n) (t) ≤ f (t,u(t),u (t), , u (n−3) (t),β (n−2) (t),β (n−1) (t)), where u ( j) (t) = δ(α ( j) (t),u ( j) (t),β ( j) (t)), j = 0,1, ,n − 3; (iii) P i (u(a), , u (i−2) (a),α (i−1) (a),α (i) (a),u (i+1) (a), , u (n−1) (a)) ≤ 0 ≤ P i (u(a), , u (i−2) (a),β (i−1) (a),β (i) (a),u (i+1) (a), , u (n−1) (a)), P n (u(b), ,u (n−3) (b),α (n−2) (b), α (n−1) (b)) ≤ 0 ≤ P n (u(b), ,u (n−3) (b),β (n−2) (b),β (n−1) (b)), where i = 1,2, , n −1, u (n−2) (a)= δ(α (n−2) (a),u (n−2) (a),β (n−2) (a)), u (n−1) (a)= δ(−N, u (n−1) (a),N). Definit ion 2.2. A continuous function f (t,ξ 0 , , ξ n−1 ) is said to satisfy a Nagumo condi- tionwithrespecttovariableξ n−1 on the set Ᏸ = t,ξ 0 , , ξ n−1 | t ∈ I; ξ j ≤ r j , j =0,1, ,n−2, r j is a positive constant; ξ n−1 ∈ R (2.2) in case there exists function Φ(t) ∈ C([0,+∞],(0,+∞)), such that f t,ξ 0 , , ξ n−1 ≤ Φ ξ n−1 , +∞ sds Φ(s) = +∞. (2.3) 2.2. The modified problem. Assume that there are two functions α(t), β(t) satisfying α ( j) (t) ≤ β ( j) (t), j = 0,1, ,n − 2. (2.4) Guangwa Wang et al. 3 We define function f t, y, y , , y (n−1) ≡ f t, y, y , , y (n−1) + h y (n−2) , (2.5) where y ( j) (t) = δ(α ( j) (t), y ( j) (t),β ( j) (t)) (j = 0,1, ,n − 2) and y (n−1) (t) = δ(−N, y (n−1) (t),N). N is a positive constant such that N>max t∈I 2M b − a , α (n−1) (t) , β (n−1) (t) , (2.6) N 2M/(b −a) sds Φ(s) > 2M, (2.7) in which M>max t∈I {|α (n−2) (t)|,|β (n−2) (t)|}. h(y (n−2) )iscontinuous,bounded,and h y (n−2) ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ < 0ify (n−2) <α (n−2) , = 0ifα (n−2) ≤ y (n−2) ≤ β (n−2) , > 0ify (n−2) >β (n−2) . (2.8) Such function h( ·)iseasytoobtain,forexample,let h y (n−2) ≡ y (n−2) − y (n−2) 1+ y (n−2) − y (n−2) . (2.9) In addition, we define P i y(t), y (t), , y (n−1) (t) ≡ δ α (i−1) (t), y (i−1) (t) − P i y(t), y (t), , y (n−1) (t) ,β (i−1) (t) , i = 1,2, ,n − 1, P n y(t), y (t), , y (n−1) (t) ≡ δ α (n−2) (t), y (n−2) (t) − P n y(t), y (t), , y (n−1) (t) ,β (n−2) (t) . (2.10) Then we consider the following modified problem: y (n) = f t, y, y , , y (n−1) , y (i−1) (a) = P i y(a), y (a), , y (n−1) (a) , i = 1, ,n − 1, y (n−2) (b) = P n y(b), y (b), , y (n−1) (b) . (2.11) 4Differential inequalities method to nth-order BVPs 2.3. Preparative theorem. Lemma 2.3. Assume that (A1) BVP (1.1)hasaboundingpair(α(t),β(t)) on the interval I by Definition 2.1; (A2) the function f (t, y, y , , y (n−1) ) in BVP (1.1) satisfies the Nagumo condition with respect to y (n−1) (t) by Definition 2.2 . Then BVP (2.11)hasasolutiony(t) ∈ C n (I,R) such that α (i) (t) ≤ y (i) (t) ≤ β (i) (t), i = 0,1, ,n − 2, y (n−1) (t) ≤ N, t ∈ I, (2.12) where N is the positive constant given in the definition of f . The proof of Lemma 2.3 is a simple consequence of the following three propositions. Proposition 2.4. The modified BVP (2.11)hasasolutiony(t) ∈ C n (I,R). Proof. Consider y (n) = λ f t, y, , y (n−1) ≡ g(t), y (i−1) (a) = λP i y(a), , y (n−1) (a) ≡ g i (a), y (n−2) (b) = λP n y(b), , y (n−1) (b) ≡ g n (b), i = 1,2, ,n − 1, (2.13) where λ ∈ [0,1]. From the representations of f , P i ,andP n ,weknowthaty (n) (t), y (i−1) (a) (i = 1,2, ,n − 1), and y (n−2) (b) all are bounded. Also, by the mean value theorem, we may ensure that y (n−1) (t), , y (t), y(t) all are bounded functions in I. In fact, by the mean value theorem, there exists some ξ ∈ (a,b) satisfying y (n−2) (b) − y (n−2) (a) = y (n−1) (ξ)(b − a), (2.14) then y (n−1) (ξ) is bounded. From y (n−1) (t) − y (n−1) (ξ) = y (n) (η)(t − ξ) ∀t ∈ [a,b], (2.15) y (n−1) (t) is bounded. Thus, from y (i) (t) − y (i) (a) = y (i+1) (ζ)(t − a), η ∈ (a,b), i = 0,1, ,n − 2, (2.16) it is easy to see that y (n−2) (t), , y (t), y(t) all are bounded in I. Let Ω ={y( t) ∈ C n (I,R) |y (i) (t) <K,forallt ∈ I, i = 0,1, ,n − 1,K is some sufficiently large positive constant }.ThenΩ is a bounded open set. BVP (2.13)canbe equivalently written as the following integral equation: y(t) = c 1 + c 2 t + c 3 t 2 + ···+ c n t n−1 + t a t n−1 a ··· t 1 b g(s)dsdt 1 ···dt n−1 ≡ T λ y, (2.17) Guangwa Wang et al. 5 where T λ is an integral operator with a parameter λ and (c 1 , , c n )isdeterminedbythe system of equations c 1 + c 2 a + c 3 a 2 + ···+ c n a n−1 = g 1 (a), c 2 + c 3 · 2a + ···+ c n (n − 1)a n−2 = g 2 (a), . . . c n−1 (n − 2)(n − 3)···3+c n (n − 1)!a = g n−1 (a), c n−1 (n − 2)(n − 3)···3+c n (n − 1)!b = g n (b) − b a t 1 b g(s)dsdt 1 . (2.18) Let H(λ, y) = (I − T λ )(y), then H : [0,1] × Ω → R n is continuous, where I is identity mapping. Let h λ (y) = H(λ, y), then 0 /∈ h λ (∂Ω). In fact, for all y ∈ ∂Ω, y ≥ K.Notic- ing that K is sufficiently large, we have h λ (y) =y − T λ y ≥ y −T λ y ≥ K−T λ y > 0 ∀λ ∈ [0,1]. (2.19) Thus, 0 / ∈ h λ (∂Ω). By the homotopy invariance theorem of topological degree, deg(h λ , Ω,0) is a constant, in particular, deg(h 1 ,Ω,0) = deg(h 0 ,Ω, 0). Noticing that 0 ∈ Ω,bythe normality of topological degree, we have deg h 1 ,Ω,0 = deg h 0 ,Ω,0 = deg I − T 0 ,Ω,0 = deg I,Ω,0 = 1. (2.20) Hence, by the solvability theorem of topological degree, it is clear that there exists some y(t) satisfying (2.17), then this proposition is proved. Proposition 2.5. Every solution y(t) of the modified BVP (2.11)satisfies α (i) (t) ≤ y (i) (t) ≤ β (i) (t), t ∈ I, i = 0,1, ,n − 2. (2.21) Proof. First, we show that α (n−2) (t) ≤ y (n−2) (t) ≤ β (n−2) (t), t ∈ I. (2.22) If α (n−2) (t) ≤ y (n−2) (t) is not true, then there exists some ξ ∈ [a,b], such that max t∈I α (n−2) (t) − y (n−2) (t) = α (n−2) (ξ) − y (n−2) (ξ) > 0. (2.23) Then ξ = a,b by the boundary conditions of BVP (2.11). Thus α (n−1) (ξ) − y (n−1) (ξ) = 0, (2.24) α (n) (ξ) − y (n) (ξ) ≤ 0. (2.25) 6Differential inequalities method to nth-order BVPs However, on the other hand, from the definition of α(t) and that y(t) is a solution of (2.11), we have α (n) (ξ) − y (n) (ξ) ≥ f ξ, y(ξ), , y (n−3) (ξ),α (n−2) (ξ),α (n−1) (ξ) − f ξ, y(ξ), , y (n−3) (ξ), y (n−2) (ξ), y (n−1) (ξ) − h y (n−2) (ξ) =− h y (n−2) (ξ) > 0. (2.26) This contradicts (2.25). Hence, α (n−2) (t) ≤ y (n−2) (t), t ∈ I. (2.27) A similar proof shows that y (n−2) (t) ≤ β (n−2) (t), t ∈ I. (2.28) To s um u p, ( 2.22)istrue.From(2.22), the function y (n−3) (t) − α (n−3) (t) is increasing in I. Noticing α (n−3) (a) ≤ y (n−3) (a), (2.29) we know that α (n−3) (t) ≤ y (n−3) (t). A similar proof shows y (n−3) (t) ≤ β (n−3) (t). Using the same argument, it follows that α (i) (t) ≤ y (i) (t) ≤ β (i) (t), i = n − 4,n− 5, ,2,1. Thus, the proof of Proposition 2.5 is completed. Proposition 2.6. For every solution y(t) of the modified BVP (2.11)holds y (n−1) (t) ≤ N, t ∈ I. (2.30) Proof. Suppose that there exists some τ ∈ [a,b]suchthat y (n−1) (τ) >N. (2.31) Without loss of generality, we assume that y (n−1) (τ) >N. There exists ξ ∈ (a,b), such that y (n−1) (ξ) = y (n−2) (b) − y (n−2) (a) b − a ≤ 2M b − a <N. (2.32) Hence, there exists some subinterval [c, d](or[d, c]) ⊂ [a,b]suchthat y (n−1) (c) = 2M b − a , y (n−1) (d) = N, 2M b − a ≤ y (n−1) (t) ≤ N, ∀t ∈ [c,d](or[d,c]). (2.33) Guangwa Wang et al. 7 From condition (A2), d c y (n−1) (s)y (n) (s) Φ y (n−1) (s) ds ≤ d c y (n−1) (s)ds = y (n−2) (d) − y (n−2) (c) ≤ 2M. (2.34) On the other hand, from (2.7)weknowthat d c y (n−1) (s)y (n) (s) Φ y (n−1) (s) ds = N 2M/(b −a) rdr Φ(r) = N 2M/(b −a) rdr Φ(r) > 2M. (2.35) This inequality contradicts the above one and Proposition 2.6 holds. 3. Main theorem Now, the main result of this paper is given in the following theorem. Theorem 3.1. Assume that the conditions (A1), (A2) in Le mma 2.3 hold and added to (A3). The function P i (η 0 , , η n−1 )(i = 1,2, ,n) satisfies (i) P i (η 0 , , η n−1 ) is increasing in η i−1 and decreasing in η i , i = 1,2, ,n − 2; (ii) P n−1 (η 0 , , η n−1 ) is decreasing in η n−1 ; (iii) P n (η 0 , , η n−1 ) is increasing in η n−1 . Then BVP (1.1)hasasolutiony(t) ∈ C n (I,R) such that α (i) (t) ≤ y (i) (t) ≤ β (i) (t), i = 0,1, ,n − 2, y (n−1) (t) ≤ N, t ∈ I, (3.1) where N is the positive constant given in the definition of f . Proof. From Lemma 2.3 and the definition of f , the solution y(t) of the modified BVP (2.11) satisfies (1.1). As soon as it is proved that y(t) satisfies the b oundary conditions of (1.1) under condition (A3), we may say that y(t) is a solution of BVP (1.1). First, we prove P i y(a), , y (n−1) (a) = 0, i = 1,2, ,n − 2. (3.2) Case 1. Suppose that α (i−1) (a) ≤ y (i−1) (a) − P i y(a), , y (n−1) (a) ≤ β (i−1) (a). (3.3) Then y (i−1) (a) = P i y(a), , y (n−1) (a) = y (i−1) (a) − P i y(a), , y (n−1) (a) . (3.4) Thus P i y(a), , y (n−1) (a) = 0. (3.5) 8Differential inequalities method to nth-order BVPs Case 2. Suppose that there exists some i ∈{1,2, ,n − 2} such that α (i−1) (a) >y (i−1) (a) − P i y(a), y (a), , y (n−1) (a) . (3.6) Then y (i−1) (a) = P i y(a), y (a), , y (n−1) (a) = α (i−1) (a). (3.7) Hence P i y(a), y (a), , y (n−1) (a) > 0. (3.8) From Propositions 2.5 and 2.6 and condition (A3), P i y(a), , y (i−2) (a),α (i−1) (a),α (i) (a), y (i+1) (a), , y (n−1) (a) > 0. (3.9) It is easy to see that the last inequality contradicts Definition 2.1(iii). Therefore, Case 2 is not true. Case 3. Suppose that there exists some i ∈{1,2, ,n − 2} such that y (i−1) (a) − P i y(a), y (a), , y (n−1) (a) >β (i−1) (a). (3.10) Then by the analogous analysis, we have P i y(a), , y (i−2) (a),β (i−1) (a),β (i) (a), y (i+1) (a), , y (n−1) (a) ≤ P i y(a), , y (n−1) (a) <0. (3.11) Obviously, the last inequality contradicts Definition 2.1(iii). Therefore, this case cannot hold. To sum u p, ( 3.2)holds. A similar proof shows that P n−1 y(a), y (a), , y (n−1) (a) = 0, P n y(b), y (b), , y (n−1) (b) = 0. (3.12) The proof is completed. 4. A generalized problem Now, we consider the following boundary value problem with more general boundary conditions: y (n) = f t, y, , y (n−1) , P i y(a), , y (n−1) (a), y(b), , y (n−1) (b) = 0, (4.1) where t ∈ I, i = 1,2, ,n, f and P i are continuous functions. Similarly to Definition 2.1, we give the following. Guangwa Wang et al. 9 Definit ion 4.1. Assume α(t),β(t) ∈ C n (I,R). The pair of functions (α(t),β(t)) is called a bounding function pair of BVP (4.1)incasethatforallu(t) ∈ C n (I,R) (i)thesameasDefinition 2.1(i); (ii) the same as Definition 2.1(ii); (iii) P i u(a), , α (i−1) (a),α (i) (a), , u (n−1) (a),u(b), ,u (n−1) (b) ≤ 0 ≤ P i u(a), , β (i−1) (a),β (i) (a), , u (n−1) (a),u(b), ,u (n−1) (b) , P n u(a), , u (n−1) (a),u(b), ,u (n−3) (b),α (n−2) (b),α (n−1) (b) ≤ 0 ≤ P n u(a), , u (n−1) (a),u(b), ,u (n−3) (b),β (n−2) (b),β (n−1) (b) , (4.2) where i = 1,2, ,n − 1. For B VP (4.1), we have the following existence theorem. Theorem 4.2. Assume that (A1) BVP (4.1)hasaboundingfunctionpair(α(t),β(t)) in the interval I by Definition 4.1; (A2) the function f (t, y, y , , y (n−1) ) in BVP (4.1) satisfies the Nagumo condition with respect to y (n−1) (t) by Definition 2.2 ; (A3) the function P i (η 0 , , η n−1 ,ζ 0 , , ζ n−1 )(i = 1,2, ,n) satisfies (i) P i (η 0 , , η n−1 ,ζ 0 , , ζ n−1 ) is increasing in η i−1 and decreasing in η i , i = 1,2, , n − 2; (ii) P n−1 (η 0 , , η n−1 ,ζ 0 , , ζ n−1 ) is decreasing in η n−1 ; (iii) P n (η 0 , , η n−1 ,ζ 0 , , ζ n−1 ) is increasing in ζ n−1 . Then BVP (4.1)hasasolutiony(t) ∈ C n (I,R) such that α (i) (t) ≤ y (i) (t) ≤ β (i) (t), i = 0,1, ,n − 2, y (n−1) (t) ≤ N, t ∈ I, (4.3) where N is the positive constant given in the definition of f . Proof. Consider the modified problem y (n) = f t, y, , y (n−1) , y (i−1) (a) = P i (a), y (n−2) (b) = P n (b), i = 1,2, ,n − 1. (4.4) The modified function f (t, y, , y (n−1) )isdefinedasBVP(2.11), and P i (t) ≡ P i y(t), , y (n−1) (t), y(b + a − t), , y (n−1) (b + a − t) ≡ δ α (i−1) (t), y (i−1) (t) − P i y(t), , y (n−1) (t), y(b + a − t), , y (n−1) (b + a − t) ,β (i−1) (t) , (4.5) 10 Differential inequalities method to nth-order BVPs where i = 1,2, ,n − 1, P n (t) ≡ P n y(b + a − t), , y (n−1) (b + a − t), y(t), , y (n−1) (t) ≡ δ α (n−2) (t), y (n−2) (t) − P n y(b + a − t), , y (n−1) (b + a − t), y(t), , y (n−1) (t) ,β (n−2) (t) . (4.6) Using the same argument as the proof of Lemma 2.3, it follows that under the conditions (A1) and (A2) ,BVP(4.4)hasasolutiony(t) satisfying the two inequalities in the con- clusions of Lemma 2.3.Furthermore,inananalogouswaytotheproofofTheorem 3.1, it fol l ows that the solution y(t)ofBVP(4.4)isasolutionofBVP(4.1). Consequently, the proof of Theorem 4.2 is completed. The details of the proof w ill be omitted. 5. An example In this section, we study an example by making use of Theorems 3.1 and 4.2. Example 5.1. Consider the 4th-order nonlinear boundary value problem y (iv) = (t − y) 2 − t 1+t 2 y + 112 sin2 1+(y ) 2 sin(y )+ t + t 2 2 1+(y ) 2 , 4y(1) − 1 8 y (1) 3 − y (1) + k 6 y(2) = A, 5y (1) − 1 2 y (1) + k 8 y (2) 2 = B, y(1) +2y (1) − y (1) − k 2 y (2) = C, ky(1) − y (2) − 4 y (2) 2 +4 y (2) 3 = D, (5.1) where t ∈ [1,2], k is a constant. Let f t,ξ 0 ,ξ 1 ,ξ 2 ,ξ 3 = t − ξ 0 2 − t 1+t 2 ξ 1 + 112 sin2 1+ξ 2 1 sinξ 2 + t + t 2 2 1+ξ 2 3 , P 1 η 0 ,η 1 ,η 2 ,η 3 ,ζ 0 ,ζ 1 ,ζ 2 ,ζ 3 = 4η 0 − 1 8 η 3 1 − η 2 + k 6 ζ 0 − A, P 2 η 0 ,η 1 ,η 2 ,η 3 ,ζ 0 ,ζ 1 ,ζ 2 ,ζ 3 = 5η 1 − 1 2 η 2 + k 8 ζ 2 1 − B, P 3 η 0 ,η 1 ,η 2 ,η 3 ,ζ 0 ,ζ 1 ,ζ 2 ,ζ 3 = η 0 +2η 2 − η 3 − k 2 ζ 2 − C, P 4 η 0 ,η 1 ,η 2 ,η 3 ,ζ 0 ,ζ 1 ,ζ 2 ,ζ 3 = kη 0 − ζ 1 − 4ζ 2 2 +4ζ 3 3 − D. (5.2) [...]... 2, 757–781 12 Differential inequalities method to nth-order BVPs [14] G Wang, M Zhou, and L Sun, Existence of solutions of two-point boundary value problems for the systems of nth-order differential equations, Journal of Nanjing University Mathematical Biquarterly 19 (2002), no 1, 68–79 (Chinese) [15] X Yang, The method of lower and upper solutions for systems of boundary value problems, Applied Mathematics... and lower solution methods for fully nonlinear boundary value problems, Journal of Differential Equations 180 (2002), no 1, 51–64 [5] L H Erbe, Existence of solutions to boundary value problems for second order differential equations, Nonlinear Analysis 6 (1982), no 11, 1155–1162 [6] Ch Fabry and P Habets, Upper and lower solutions for second-order boundary value problems with nonlinear boundary conditions,... 861–866 [11] M A O’Donnell, Semi-linear systems of boundary value problems, SIAM Journal on Mathematical Analysis 15 (1984), no 2, 316–332 [12] M H Pei, Nonlinear two-point boundary value problems for nth-order nonlinear differential equations, Acta Mathematica Sinica 43 (2000), no 5, 921–930 [13] F Sadyrbaev, Nonlinear fourth-order two-point boundary value problems, The Rocky Mountain Journal of Mathematics... (1986), no 10, 985–1007 [7] F A Howes, Differential inequalities and applications to nonlinear singular perturbation problems, Journal of Differential Equations 20 (1976), no 1, 133–149 [8] W G Kelley, Some existence theorems for nth-order boundary value problems, Journal of Differential Equations 18 (1975), no 1, 158–169 [9] Z Lin and M Zhou, Perturbation Methods in Applied Mathematics, Jiangsu Education... −2], it is easy to prove that (α(t),β(t)) is a bounding pair of BVP (5.1) and all assumptions of Theorems 3.1 and 4.2 are fulfilled, respectively Hence, for any of the two cases, BVP (5.1) has at least one solution y(t) satisfying −t 2 ≤ y(t) ≤ t, −2t ≤ y (t) ≤ 1, −2 ≤ y (t) ≤ 0, t ∈ [1,2] (5.4) Acknowledgments Sincere thanks to the anonymous Referee for his/her careful reading and the Editors for their... equations with two point boundary conditions, Journal of Mathematical Analysis and Applications 285 (2003), no 1, 174–190 [2] K W Chang and F A Howes, Nonlinear Singular Perturbation Phenomena: Theory and Applications, Applied Mathematical Sciences, vol 56, Springer, New York, 1984 [3] Z Du, W Ge, and X Lin, Existence of solutions for a class of third-order nonlinear boundary value problems, Journal . DIFFERENTIAL INEQUALITIES METHOD TO nTh-ORDER BOUNDARY VALUE PROBLEMS GUANGWA WANG, MINGRU ZHOU, AND LI SUN Received 31 March 2005; Revised 18 October 2005; Accepted 7 December. Corporation Journal of Inequalities and Applications Volume 2006, Article ID 12040, Pages 1–12 DOI 10.1155/JIA/2006/12040 2Differential inequalities method to nth-order BVPs clearer compared with the method of. Nonlinear fourth-order two-point boundary value problems, The Rocky Mountain Journal of Mathematics 25 (1995), no. 2, 757–781. 12 Differential inequalities method to nth-order BVPs [14] G. Wang, M.